论文标题
在一个简单模型中的动力混乱
Dynamical Chaos in a Simple Model of a Knuckleball
论文作者
论文摘要
指关节球也许是棒球中最神秘的球场。与其他棒球球场（例如快球或曲线球）相比，依靠球表面上凸起的接缝存在以创造不对称的流程，这一轨迹非常具有挑战性。以前对大量关节球的实验跟踪表明，它们可以基本上朝任何方向移动，而相对于仅拖动轨迹的期望。这导致人们猜测指关节表现出混乱的运动。在这里，我们开发了一个相对简单的指关节模型，其中包括从不对称流程中进行的二次阻力和提升，这是从缓慢旋转棒球的实验测量中取出的。相比之下，我们的模型确实可以表现出动态混乱，而在飞行中省略球上的扭矩的模型不会显示出混乱的行为。表明，指关节球的相位空间位置的不确定性在从投手到本垒板上的球飞机上的$ 10^6 $中增长。我们量化了模型参数对模型中实现的混乱的影响，特别表明，最大Lyapunov指数与扭矩有效杆的平方根大致成正比，并且与音高的初始速度大致成比例。我们证明了分叉的存在，这些分叉在球到达板的位置可能会发生变化，而该板的板对于特定的初始条件的特定初始条件类似于专业的敲打球员使用的特定初始条件。当我们以经验不对称力测量的更忠实表示的形式引入额外的复杂性时，我们发现可能的初始条件的较大部分会导致动态混乱。
The knuckleball is perhaps the most enigmatic pitch in baseball. Relying on the presence of raised seams on the surface of the ball to create asymmetric flow, a knuckleball's trajectory has proven very challenging to predict compared to other baseball pitches, such as fastballs or curveballs. Previous experimental tracking of large numbers of knuckleballs has shown that they can move in essentially any direction relative to what would be expected from a drag-only trajectory. This has led to speculation that knuckleballs exhibit chaotic motion. Here we develop a relatively simple model of a knuckleball that includes quadratic drag and lift from asymmetric flow which is taken from experimental measurements of slowly rotating baseballs. Our models can indeed exhibit dynamical chaos as long In contrast, models that omit torques on the ball in flight do not show chaotic behavior. Uncertainties in the phase space position of the knuckleball are shown to grow by factors as large as $10^6$ over the flight of the ball from the pitcher to home plate. We quantify the impact of our model parameters on the chaos realized in our models, specifically showing that maximum Lyapunov exponent is roughly proportional to the square root of the effective lever arm of the torque, and also roughly proportional to the initial velocity of the pitch. We demonstrate the existence of bifurcations that can produce changes in the location of the ball when it reaches the plate of as much as 1.2 m for specific initial conditions similar to those used by professional knuckleball pitchers. As we introduce additional complexity in the form of more faithful representations of the empirical asymmetry force measurements, we find that a larger fraction of the possible initial conditions result in dynamical chaos.
